How do we know we have all subsets of a set?

Question

The axiom schema of comprehension basically says that every definable subclass of a set is also a set. However, we know there are only countably many formulas in the language of set theory. So, this axiom seems to only give us countably many subsets of a set. Intuitively, this seems like a very bad result since we know there should be way more subsets of large cardinalities than this. Intuitively, I want ALL subsets of a set to exist, regardless of whether or not they are nameable by some formula. So, my main first question is how to alleviate the worry that we are missing on lots of different subsets of a set, and therefore our math may think the power set of a set is way smaller than it should be. My second side question is that given that the axiom of comprehension only seems to give us countably many subsets of a set, where do all the other subsets come from? We can prove that the powerset of the natural numbers is not countable, and of course I understand that formal proof. Maybe there is nothing more to say about this than just pointing at the proof, but some explanation as to what generates these subsets, given that the real thing that we want generating these subsets (namely their mere existence rather than definability) is not generating these subsets.